Disney's Pop Century Resort has an address of 1050 Century Drive. According to the Orange County Property Appraiser, Century Drive is the road from Victory Way to the parking lots, but no signs give the name. The only street signs, overhead internally-lit ones at the Victory Way traffic light, say "Disney's Pop Century". Someone following directions to turn at Century Drive may miss their turn.

My first inclination is to move the name to an unsigned_name field, but would this break anything such as address searching?

A more complicated case: the entrance to Coronado Springs has a name that is signed internally, but the signs at the entrance instead say "Coronado Springs".

I think your proposed change would break Nominatim’s TIGER-based address interpolation, but if OSM has address data for the entire street (either explicitly or using interpolation ways), I don’t think it’ll be a problem at all. Then again, since “Century Drive” is in actual use – not just in the county database – maybe alt_name would be appropriate?

It sounds silly, but maybe you could split the street leading into Coronado Springs, so that the segment from the big intersection up to the first sign or gate is name=Coronado Springs and unsigned_name=Avenida del Centro, and the rest is name=Avenida del Centro.

Vanity street names can lead to all sorts of weirdness. I can’t remember the names exactly, but consider a Ye Olde Drive-In Theater off Main Street (with a Main Street address) and the blade signs at the Main Street entrance that say “Movies →”. Or consider Loveland High School, whose driveway has a vanity name of “Tiger Trail”. The school has two addresses: “1 Tiger Drive” and “[some four-digit number] Rich Road”. Thankfully, I can’t remember that four-digit number. :^)

I know, but it sounds like the road is signed as if it’s named “Disney’s Pop Century”. My suggestion above optimizes for drivers reading printouts, but there doesn’t really seem to be an ideal solution without using some obscure tagging scheme for the signs themselves.